Collaborative Filtering: A Comparison of Graph-Based Semi-Supervised Learning Methods and Memory-Based Methods

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1 70 Computer Science 8 Collaborative Filtering: A Comparison of Graph-Based Semi-Supervised Learning Methods and Memory-Based Methods Rasna R. Walia Collaborative filtering is a method of making predictions about the interests of a user based on interest similarity to other users and consequently recommending the predicted items. There is a widespread use of collaborative filtering systems in commercial websites, such as Amazon.com, which has popularized item-based methods. There are also many music and video sites such as ilike and Everyone s a Critic (EaC that implement collaborative filtering systems. This trend is growing in product-based sites. This paper discusses the implementation of graph-based semisupervised learning methods and memory-based methods to the collaborative filtering scenario and compares these methods to baseline methods such as techniques based on weighted average. This work compares the predictive accuracy of these methods on the MovieLens data set. The metrics used for evaluation measure the accuracy of generated predictions based on already known, held-out ratings that constitute the test set. Preliminary results indicate that graph-based semi-supervised learning methods perform better than baseline methods. However, some of the memory-based methods outperform the graph-based semi-supervised learning methods as well as the baseline methods.. Introduction Collaborative filtering is basically a method of matching people with similar interests for the purpose of making recommendations. The basic assumption of collaborative filtering is that those who agreed in the past tend to agree in the future. An example of a collaborative filtering system in use is Amazon.com, where new books are recommended to users based on what they have previously bought as well as their similarity to other users. The task of collaborative filtering is split into prediction and recommendation. Collaborative prediction refers to the task of predicting preferences of users based on their preferences so far, and how they relate to the preferences of other users. On the other hand, collaborative recommendation is the task of specifying a set of items that a user might like or find useful. In this work, focus was on the collaborative prediction task and the testing of different algorithms for their accuracy. Collaborative filtering systems are also distinguished based on whether they use implicit or explicit votes. Explicit voting refers to a user expressing his preference 70

2 Collaborative Filtering 7 for an item, usually on a discrete numerical scale. On the other hand, implicit voting arises from interpreting user behaviour to estimate a vote or preference. Implicit votes are often based on things like browsing data, purchase history or other types of information access patterns. Throughout this work, explicit votes were used to form predictions, because of their simplicity and natural interpretation. Collaborative filtering algorithms suggest new items or predict their usefulness for a particular user based on the user s previous likings and the opinions of other like-minded users. Typically, there is a list of m users U = {u, u,, u m } and a list of n items I = {i, i,, i n }. Each user u b is associated with a list of items I ub on which he has expressed a liking, between a range of 0-5, with meaning least preferred, 5 meaning most preferred and 0 meaning the item has not been rated by the user. The entire m x n user-item data is represented as a ratings matrix, R. Each entry r i, in R represents the rating of the i th user on the th item. An example ratings matrix with 5 users and 0 movies is given below: This paper contains three contributions: The application of graph-based semi-supervised learning methods within the collaborative filtering domain. Implementation of several methods on a non-trivial sized data set i.e. the MovieLens data set, with 00,000 ratings from 943 users on,68 movies. Alteration to memory-based methods: User-based methods were applied to items and a significant improvement in the accuracy of the predictions was observed. This paper is organized as follows: Section briefly discusses related work. Section 3 outlines the experimental methods employed, including a description of the data set and error measures used. The results are presented in Section 4. Section 5 presents a discussion of the results obtained and Section 6 outlines avenues for future work.. Related Work A number of collaborative filtering techniques have been proposed, of which the most popular ones are those based on the correlation criteria and matrix factorization. 7

3 7 Computer Science Breese et al. [998] carried out an empirical analysis of the predictive accuracy of several memory-based algorithms like techniques based on the correlation coefficient and vector-based similarity as well as model-based methods like the Naïve Bayes formulation and Bayesian Networks. They evaluated their algorithms on the MS Web data set, Neilsen data set and EachMovie data set. A key observation from their work is that the performance of the different methods is greatly influenced by the nature of the data set and the availability of votes with which to make predictions. Results generated by memory-based methods are generally quite accurate. A maor drawback of these methods is that they are computationally very expensive because the similarity between each pair of users needs to be computed before predictions can be made on the desired items. These algorithms also cannot detect cases of item synonymy. Another maor disadvantage of memory-based methods is that they do not construct any explicit statistical models so nothing is really ``learnt from the available user profiles. A maor computation in memory-based methods is the calculation of similarities between users. Algorithms where predictions are made based on the similarity computation among items have been implemented by Sarwar et al. [00]. These methods are known as item-based collaborative filtering algorithms. Unlike the user-based methods, the item-based approach looks into the set of items the target user has rated and computes how similar they are to the target item i and then selects k most similar items. Once the most similar items have been found, the prediction is computed by taking a weighted average of the target user s ratings on the similar items. In cases where large datasets are being worked on, these methods provide better quality recommendations than user-based methods [Sarwar et al. 00]. Item-based methods allow similarity computations among items to be performed in advance, leading to faster recommendations for a user. In many cases, the collaborative prediction task is viewed as a classification problem [Marlin 004]. Algorithms like the k-nearest neighbour classifier among others have been implemented in Marlin s work. Semi-supervised learning is also considered a classification method. It targets situations where labelled data are scarce and unlabelled data are in abundance. Semi-supervised learning on a graph has been studied from different perspectives in Belkin and Niyogi [004], Herbster et al. [005] and Herbster and Pontil [006]. A common theme in all these papers is the use of the graph Laplacian, L. One of the characteristics of the data used within the collaborative filtering framework is sparsity. The user-movie matrix, where rows represent the users and columns represent the movies, has relatively few actual ratings and has many missing values which are represented as zeros. Dimensionality reduction methods and matrix decomposition techniques such as singular value decomposition (SVD based prediction algorithms can overcome the sparsity problem by utilizing the latent relationships that are captured in the low-rank approximations of the usermovie matrix. [Sarwar et al. 000], [Kleeman et al. 006]

4 Collaborative Filtering 73 Several model-based methods such as Bayesian networks [Breese et al. 998] and clustering [Connor and Herlocker 00], [George and Merugu 005] have been applied to the collaborative filtering domain. These different methods provide item recommendations by first building a model of user ratings. The Bayesian network model formulates a probabilistic model for the collaborative filtering problem while the clustering model treats it as a classification problem. The clustering model works by similar users and/or similar items in the same class and estimating the probability that a particular user belongs to certain class C and from there, computes the conditional probability of ratings. In this work, graph-based semi-supervised learning algorithms and memorybased algorithms have been implemented and compared with each other because they work in a similar manner. Both methods operate over the entire user database to generate the required prediction for a particular movie for the given active user.. Experimental Method. Methods Employed There are generally two broad classes of collaborative filtering algorithms. Memorybased algorithms operate over the entire user database to make predictions while model-based algorithms use the user database to learn a model which is then used for prediction. The emphasis of this work is on memory-based methods. Semi-supervised learning is a class of machine learning techniques that makes use of both labelled and unlabelled data for training. Typically, such learning problems are characterized by a small amount of labelled data with a large amount of unlabelled data. It has been found that the use of labelled data in conunction with unlabelled data can produce considerable improvement in learning accuracy. The graph-based learning algorithms presented in this paper are all categorized as semi-supervised learning methods. The problem of collaborative filtering lends itself well to the semi-supervised learning framework since an analogy can be drawn between labelled data and user ratings for movies as well as the unknown ratings and unlabelled data. 3.. Graph-Based Semi-Supervised Learning Algorithms Two graph-based semi-supervised learning algorithms were used in this work: Minimum Norm Interpolation: This method uses a Laplacian kernel to predict labels on the unlabelled data. Harmonic Energy Minimizing Functions: This method uses the graph Laplacian and labels on the labelled data to predict the labels on the unlabelled data.

5 74 Computer Science 3... Hilbert Space of Functions on a Graph Functions defined on the graph are represented by a Hilbert space associated with the graph Laplacian. Let G = (V,E be an undirected graph with a vertex set V = {,...,n}, edge set E( G = E {( i, : i < } i, V and n x n adacency matrix A = ( A i, : i, V such that A i, = A, i = if ( i, E and zero otherwise. The graph Laplacian L is the n x n matrix defined as L = D - A, where D = diag(d,,d n and d i is the degree of vertex i. There are l labelled points and u unlabelled points. Usually, l << u and n = l + u is the total number of points in the graph Minimum Norm Interpolation Let R(G be the linear space of real-valued functions defined as g = (g,...,g n T, where ``T denotes transposition. A linear subspace H(G of R(G is defined which is orthogonal to the eigenvectors of L with zero eigenvalue, that is, H(G = { g : g T ui = 0, i =,..., r} Since G is connected L has only one eigenvector with eigenvalue zero (the constant vector and therefore n Equation : H(G = { g : g i = 0} i= Within this framework, the aim is to learn a classification function g H(G on the basis of a set of labelled vertices. g is obtained as the minimal norm interpolant in H(G to the labelled vertices, i.e., the unique solution to the problem: Equation : min { g H ( G g : g i = yi, i =,..., l} The reproducing kernel of H(G is the pseudo-inverse of the Laplacian K = L +. With the representer theorem, the coordinates of g are expressed as: Equation 3: g i = l = K i c The solution of Equation 3 is given by c = K ~ l y where K = ( K i i, =. In this work, the following symmetric positive definite graph kernel has been used: b K = + T c L + b + ci, (0 < b,0 c This method assumes that all labels are centred around zero and hence, all ratings in the data set are shifted relative to one of the following: User average, Movie average, Weighted average, User median, Movie median Harmonic Energy Minimizing Functions There are l labelled points and u unlabelled points. L denotes the labelled set and U denotes the unlabelled set of points. In this formulation, the labels on the labelled + ~

6 Collaborative Filtering 75 data lie between two intervals a and b such that y i = y L(i [a,b] for i =,,l. This constraint is denoted by y L=yl. To satisfy the requirement that unlabelled points that are nearby in the graph have similar labels, the energy is defined to be: Equation 4: E(y = i, E ( G ( y i y so that low energy corresponds to a slowly varying function over the graph. The graph Laplacian matrix and known labels are used to calculate the labels of unlabelled data points. The harmonic property means that the value at each unlabelled node is the average of the neighbouring nodes. The harmonic energy minimizing function f is computed with matrix methods. The Laplacian matrix L is partitioned into blocks for labelled and unlabelled nodes as follows: Lll Llu L = Lul Lu f l f = Let f u where f l = y L and f u denotes the mean values on the unlabelled data points. The solution is given by: Equation 5: f u = L u Lul f l This method also requires labels to be centered on zero and so ratings are shifted in the user-movie matrix relative to one of the following: User average, Movie average, Weighted average, User median, Movie median. The basic idea in both the methods is that the learnt function should be smooth with respect to the graph. The smoothness in the minimum norm interpolation method and harmonic energy minimizing functions method is ensured by Equation and Equation 4, respectively. 3.. Memory-Based Algorithms Memory-based algorithms operate over the entire user database to make predictions. Given an unknown test rating (of a test item by a test user to be estimated, memory-based collaborative filtering methods first measure similarities between the test user and other user user-based methods. After that, the unknown rating is predicted by taking a weighted average of the known ratings of the test item by similar users. In this work, different user-based methods were implemented, as described by Breese et al. [998] and they are distinguished mainly by the method used to calculate the ``weight. Prediction Computation The basic task in collaborative filtering is to predict the votes of a particular user, the active user, from a database of user votes, by finding other users similar to the

7 76 Computer Science active user. The user database consists of a set of votes v b, corresponding to the vote for user b on movie. I b is the set of movies on which user b has voted. The mean vote of user b is defined as: vb = vb, I The votes of the active user, a, are predicted based on some partial information regarding the active user and a set of weights calculated from the user database. To calculate the predicted vote of the active user for item, a weighted sum of votes of the other users in the database is used. b I b Equation 6: p n a, = va + ê w(a,b(vb, vb b= where n is the number of users in the collaborative filtering database with non-zero weights and κ is a normalizing factor such that the absolute values of the weights sum to unity. w(a,b is either the distance, correlation or similarity between each user b and the active user a. An important term in Equation 6 is w(a,b, leading to the generation of different predictions depending on how it is calculated. The table below summarizes different methods used for calculating w(a,b. Table : Different Weight Calculating Methods w(a,b Distance Correlation Weight/Similarity Method Euclidean, Manhattan Pearson Correlation Vector Similarity, Default Voting, Inverse User Frequency In this work, the correlation and weight/similarity methods were used to calculate w(a,b Pearson Correlation The correlation between users a and b is: w(a,b = (v (v a, a, where the summations over are over the movies for which both users a and b have recorded votes. v v a a (v (v b, b, v b v b

8 3...4 Vector Similarity Collaborative Filtering 77 In this formulation, each user is treated as a vector of votes in n-dimensional space, where n is the number of votes in the user database. The weight between two users is computed by calculating the cosine of the angle formed by the vectors. The weight is now calculated as: v v w(a,b = where the squared terms in the denominator serve to normalize votes so that users that vote on movies will not a priori be more similar to other users. Again, the summations over are over the items for which both users a and b have recorded votes Default Voting Default voting is an extension to the correlation algorithm. Correlation, as a similarity measurement, does not work very well on sparse data sets. When two users have few movies in common, their weights tend to be over-emphasized. Default voting deals with this problem by adding a number of imaginary items that both have rated in common in order to smooth the votes. A default vote value d is assumed as a vote for movies which do not have explicit votes. The same default vote value d is taken for some number of additional items k that neither user has voted on. This has the effect of assuming that there are some additional number of unspecified movies that neither user has voted on, but they would generally agree on. In most cases, the value of d is selected to reflect a neutral or somewhat negative preference for the unobserved movies. The weight is now calculated as: (n + k( v v + kd ( v + kd ( v + kd w(a,b = ((n + k( v a, + kd ( a, b, v a, a, k I a + kd v a, k b, k Ib a, v ((n + k( b, k v b, b, + kd ( where the summations are now over the union of items that either user a or b has voted on Extension I I and n = I I. a b a b In this work, user-based methods were also applied to items. The user-based methods were modified to replace values pertaining to users with corresponding movie values. These methods are referred to as user-based methods on items in this work. To predict the vote for user a on item Equation 6 is modified slightly to become: Equation 7: p n a, = v + ê w(,i(va, i vi i= v b, + kd

9 78 Computer Science where n is the number of items in the collaborative filtering database with nonzero weights and κ is a normalizing factor such that the absolute values of the weights sum to unity. w(,i is either the distance, correlation or similarity between each item i and the active item. Two user similarity computation methods, vector similarity and default voting, were employed to calculate the similarities between movies Baseline Methods In order to determine the relative performance of the various algorithms implemented for the collaborative filtering problem in this work, the most basic prediction methods were used. These methods involve predicting an unknown rating by returning one of the following statistics as the predicted value: (a User Average; (b Movie Average; (c Weighted Average. 3. Data The different methods were evaluated on the MovieLens dataset. The fastest of methods generally ran for ten minutes on this dataset and the slowest took up to thirty minutes. The dataset consists of 00,000 ratings (in the range of - 5 from 943 users on,68 movies. A null vote i.e. a zero entry for a movie, means that the movie has not been watched by the user. Each user has rated at least 0 movies. Previous work carried out on the MovieLens dataset in relation to collaborative filtering includes: Singular value decomposition based prediction by Sarwar et al. [000]. Item-based collaborative filtering methods by Sarwar et al. [00]. Different matrix factorization techniques such as maximum margin matrix factorization, incremental SVD and repeated matrix reconstruction by Kleeman et al. [006]. This work presents a novel application of graph-based semi-supervised learning methods in the collaborative filtering domain. A similar graph-based semisupervised approach has been used to address the sentiment analysis task of rating inference for unlabelled documents. [Goldberg and Merugu 005] 3.3 Error Measures Statistical accuracy metrics were used to evaluate the quality of predictions of the different algorithms that were implemented. Also known as predictive accuracy metrics, they measure how close the predicted ratings are to the true user ratings in the test set Mean Absolute Error Mean absolute error (MAE measures the average absolute deviation between a predicted rating and the user s true rating. If the number of predicted votes in the test set for the active user is m a, then the mean absolute error for the user is:

10 Collaborative Filtering 79 S a = pa, va, ma P a where p a, is the predicted rating for user a on movie and v a, is the actual rating of user a on movie. These scores are then averaged over all the users in the test set. The lower the MAE, the more accurate is the prediction Root Mean Squared Error Root mean squared error (RMSE is a slight variation to the MAE. It squares the error before summing it up. This results in more emphasis on large errors. RMSE = where e is the error at each point and N is the number of points tested. The lower the RMSE, the more accurate is the prediction. 3.4 Experimental Protocol Experiments using the different methods were carried out in a similar manner. The data sets used in the experiments were partitioned into training sets and test sets. The methods were trained on the training sets and their performance evaluated on the test sets. In all cases, error rates taken over the set of held-out ratings used for testing and not the set of observed ratings used for training are reported. The RMSE and MAE values presented in the experiments are average error rates across multiple test sets. 4. Results To compare the predictive accuracy of the different methods, two large datasets provided by MovieLens were used. One pair is ua.base and ua.test and the other pair is ub.base and ub.test. These datasets split the main dataset into a training and test set with exactly 0 ratings per user in the test set. The sets ua.test and ub.test are disoint. The training sets (*.base have 90,570 ratings and the test sets (*.test have 9,430 ratings. The results presented in Table are for the following methods: Baseline Methods: User Average, Movie Average, Weighted Average Standard Methods - User-Based Methods: Pearson Correlation, Vector Similarity and Default Voting Variation of Standard Methods - User-Based Methods on Items: Vector Similarity and Default Voting Graph-based Semi-supervised Learning Methods: Minimum Norm Interpolation and Harmonic Energy Minimizing Functions N e e

11 80 Computer Science Table : Predictive Accuracy of the Different Methods Data Set Methods RMSE MAE ua.test ub.test ua.test ub.test User Average Movie Average Weighted Average Pearson Correlation Vector Similarity Default Voting Vector Similarity (on items Default Voting (on items Minimum Norm Interpolation Harmonic Energy Minimizing Functions Discussion The MovieLens data set used in the experiments contains 00,000 ratings from 943 users on,68 movies. Out of a possible,586,6 ratings, only 00,000 ratings are present in the data set. Due to the sparse nature of the data set, it was expected that the graph-based semi-supervised learning methods would perform better than the other methods employed. The reason for this was that these graph-based semisupervised learning methods use both labelled and unlabelled data to build better classifiers. However, the observation was that the performance of the graph-based semi-supervised learning methods were almost the same as that of the memorybased methods. The performance of the minimum norm interpolation method is approximately the same on both datasets. The harmonic energy minimizing functions method returns a slightly higher rate of error than the minimum norm interpolation method. Promising results were achieved when applying user-based methods to items. Baseline methods, such as using the weighted average as the predicted value, gave fairly good predictions. Generally, the graph-based semi-supervised learning methods perform better than the baseline methods. However, memory-based methods such as those that use Pearson correlation, vector similarity and default voting to calculate weights and generate predictions performed better than the graph-based methods. 6. Future Work There is a growing need for the use of collaborative filtering systems because of the increasing volume of customer data available on the Web and the growth of product-based websites. New technologies are continuously being exploited to improve the predictive accuracy of collaborative filtering systems.

12 Collaborative Filtering 8 This study shows that graph-based semi-supervised learning methods is one of the techniques that can be used in the collaborative filtering domain. Further research is needed to understand under what conditions certain methods work well and others do not. Future work includes implementation and observation of the performance of the methods presented in this paper to larger datasets, such as the Netflix dataset, which contains over 00 million ratings from approximately 480,000 users on 7,700 movies. Further analysis and experiments also need to be carried out to look for ways to improve the performance of the various methods presented in this paper. In addition to the above, a comparative study of the methods presented in this paper with other methods such as singular value decomposition, non-negative matrix factorization, clustering algorithms, Naive Bayes classifier and Bayesian networks needs to be carried out. It is believed that these model-based methods actually ``learn something from user profiles and item characteristics and are therefore expected to give better prediction accuracy than memory-based methods. Acknowledgements Sincere thanks to Dr. Mark Herbster (m.herbster@cs.ucl.ac.uk of the Department of Computer Science, University College London, for introducing me to the field of collaborative filtering and developing my interest in it. I would also like to thank the GroupLens Research Proect at the University of Minnesota for making the MovieLens dataset available for use. References BELKIN, M. AND NIYOGI, P. ]004]. Semi-Supervised Learning on Riemannian Manifolds. In Machine Learning, 56, pages BREESE, J.S., HECKERMAN, D. AND KADIE, C Empirical Analysis of Predictive Algorithms for Collaborative Filtering. In Proceedings of the Fourteenth Annual Conference on Uncertainty in Artificial Intelligence, pages 43 5, July 998. CONNOR, M. AND HERLOCKER, J. [00]. Clustering Items for Collaborative Filtering. In SIGIR-00 Workshop on Recommender Systems. GEORGE, T. AND MERUGU, S. [005]. A Scalable Collaborative Filtering Framework Based on Co-Clustering. In Proceedings of the Fifth IEEE International Conference on Data Mining, pages GOLDBERG, A.B. AND ZHU, X. [006]. Seeing stars when there aren t many stars: Graph-based Semi-supervised Learning for Sentiment Categorization. HLT-NAACL 006 Workshop on Textgraphs: Graph-based Algorithms for Natural Language Processing. HERBSTER, M., PONTIL, M. AND WAINER, L. [005]. Online Learning over Graphs. In ICML 005, pages HERBSTER, M. AND PONTIL, M. [006]. Prediction on a Graph with a Perceptron. In NIPS 006.

13 8 Computer Science KLEEMAN, A., HENDERSEN, N. AND DENUIT, S. Matrix Factorization for Collaborative Prediction. ICME. MARLIN, B. [004]. Collaborative Filtering: A Machine Learning Perspective. Master s Thesis, University of Toronto. SARWAR, B.M., KARYPIS, G., KONSTAN, J.A. AND RIEDL, J.T.[ 000]. Application of Dimensionality Reduction in Recommender Systems A Case Study. In ACM WebKDD Workshop, 000. SARWAR, B.M., KARYPIS, G., KONSTAN, J.A. AND RIEDL, J.T. [00]. Item-Based Collaborative Filtering Recommendation Algorithms. In Proceedings of the Tenth International World Wide Web Conference, pages 85 95, May 00. ZHU, X., LAFFERTY, J. AND GHAHRAMANI, Z. [003]. Combining Active Learning and Semi-Supervised Learning using Gaussian Fields and Harmonic Functions. In ICML, 003.